Forward Rank-Dependent Performance Criteria: Time-Consistent Investment Under Probability Distortion
aa r X i v : . [ q -f i n . M F ] A p r Forward Rank-Dependent Performance Criteria:Time-Consistent Investment Under Probability Distortion
Xue Dong He ∗ Moris S. Strub † Thaleia Zariphopoulou ‡ April 4, 2019
Abstract
We introduce the concept of forward rank-dependent performance processes, extendingthe original notion to forward criteria that incorporate probability distortions. A funda-mental challenge is how to reconcile the time-consistent nature of forward performancecriteria with the time-inconsistency stemming from probability distortions. For this, wefirst propose two distinct definitions, one based on the preservation of performance valueand the other on the time-consistency of policies and, in turn, establish their equivalence.We then fully characterize the viable class of probability distortion processes, providinga bifurcation-type result. Specifically, it is either the case that the probability distortionsare degenerate in the sense that the investor would never invest in the risky assets, orthe marginal probability distortion equals to a normalized power of the quantile functionof the pricing kernel. We also characterize the optimal wealth process, whose structuremotivates the introduction of a new, distorted measure and a related market. We thenbuild a striking correspondence between the forward rank-dependent criteria in the orig-inal market and forward criteria without probability distortions in the auxiliary market.This connection also provides a direct construction method for forward rank-dependentcriteria. A byproduct of our work are some new results on the so-called dynamic utilitiesand on time-inconsistent problems in the classical (backward) setting.
Keywords: forward criteria, rank-dependent utility, probability distortion, time-consistency,portfolio selection, inverse problems ∗ Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong.Email: [email protected] † Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong.Email: [email protected] ‡ Departments of Mathematics and IROM, The University of Texas at Austin and the Oxford-Man Institute,University of Oxford. Email: [email protected] e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria
In the classical expected utility framework, there are three fundamental modeling ingredients,namely, the model, the trading horizon and the risk preferences, and all are chosen at initialtime. Furthermore, both the horizon and the risk preferences are set exogenously to the mar-ket. In most cases, the Dynamic Programming Principle (DPP) holds and provides a backwardconstruction of the solution. This yields time-consistency of the optimal policies and an intu-itively pleasing interpretation of the value function as the intermediate indirect utility. Thereare, however, several limitations with this setting.It is rarely the case that the model is fully known at initial time. Model mis-specification andmodel decay occur frequently, especially as the investment time increases. Even if a family ofmodels is assumed, instead of a single model, and robust control criteria are used, still the initialchoice of this family of models could turn out to be inaccurate quite fast. This is also the casewhen filtering is incorporated, as it is based on the dynamics of the observation process which,however, can be wrongly pre-chosen. In addition, the trading horizon is almost never fixed,not even fully known at the beginning of an investment period. It may change, depending onupcoming (even unforeseen) opportunities and/or changes of risk preferences. Finally, it mightbe difficult to justify that one knows his utility far ahead in the future. It is more natural toknow how one feels towards uncertainty for the immediate future, rather than for instances inthe distant one (see, for example, the old note of Fischer Black, Black (1968)).Some of these limitations have been successfully addressed. For example, dynamic modelcorrection is central in adaptive control where the model is revised as soon as new incominginformation arrives and, in turn, optimization starts anew for the remaining of the horizon.Flexibility with regards to horizons has been incorporated by allowing for rolling horizons fromone (pre-specified horizon end to the next). Risk preferences have been also considered in morecomplex settings like recursive utilities, which are modeled through a “utility generator” thatdictates a more sophisticated backward evolution structure.Nevertheless, several questions related to genuinely dynamic revision of preferences andof the model, time-consistency across interlinked investment periods as well as under modelrevisions, endogenous versus exogenous specification of modeling ingredients, and others remainopen. A complementary approach that seems to accommodate some of the above shortcomings isbased on the so-called forward performance criteria. These criteria are progressively measurableprocesses that, compiled with the state processes along admissible controls, remain super-martingales and become martingales at candidate optimal policies. In essence, forward criteriaare created by imposing the DPP forward, and not backwards, in time. As a result, theyadapt to the changing market conditions, do not rely on an a priori specification of the fullmodel, and accommodate dynamically changing horizons. They produce endogenously a familyof risk preferences that follow the market in “real-time” and, by construction, preserve time-2 e, Strub and Zariphopoulou:
Forward Rank-Dependent Performance Criteria consistency across all times.Forward criteria were introduced by Musiela and Zariphopoulou (2006, 2008, 2009, 2010a,b,2011) and, further studied, among others, in ˇZitkovi´c (2009), Zariphopoulou and ˇZitkovi´c (2010),El Karoui and Mohamed (2013), Bernard and Kwak (2016), Shkolnikov et al. (2016),Choulli and Ma (2017), Liang and Zariphopoulou (2017) and Chong et al. (2018). More re-cently, they have also been considered in discrete-time by Angoshtari et al. (2019) andStrub and Zhou (2018), applied to problems arising in insurance by Chong (2018), extended tosettings with model ambiguity in K¨allblad et al. (2018) and Chong and Liang (2018), and tooptimal contract theory Nadtochiy and Zariphopoulou (2018).The associated optimization problems are ill-posed, as one specifies the initial conditionand solves the problem forward in time. In Ito markets, a stochastic PDE for the forwardcriterion was derived in Musiela and Zariphopoulou (2010b), while in Liang and Zariphopoulou(2017) a connection between forward homothetic processes, ergodic control and ergodic BSDEwas established. Other developments related to multi-scale ill-posed HJB equations and toentropic risk measures can be found in the aforementioned papers. We note that the discretecase developed in Angoshtari et al. (2019) and further studied in Strub and Zhou (2018) isparticularly challenging as there is no infinitesimal stochastic calculus and, furthermore, thereare no general results for the functional equations therein.Despite the various technical difficulties, the concept of forward performance criteria iswell defined for stochastic optimization settings whose classical (backward) analogues satisfythe DPP, and thus the martingale/supermartingale properties as well as time-consistency hold.However, these fundamentally interlinked connections break down when the backward problemsare time-inconsistent.Time-inconsistency is an important feature that arises in a plethora of interesting problemsin classical and behavioral finance. Among others, it is present in mean-variance optimization,hyperbolic discounting, and risk preferences involving probability distortions. Given, from theone hand, the recent developments in forward performance criteria and, from the other, theimportance of time-inconsistent problems, an interesting question thus arises, namely, whetherand how one can develop the concept of forward performance criteria for such settings. Herein,we study this question in the realm of rank-dependent utilities.Rank-dependent utility (RDU) theory was developed by Quiggin (1982, 1993), see alsoSchmeidler (1989), and constitutes one of the most important alternative theories of choiceunder risk to the expected utility paradigm. It features two main components: a concaveutility function that ranks outcomes and a probability distortion function. Rank-dependentutility theory is able to explain a number of empirical phenomena such as the Allais para-dox, the simultaneous investment in well-diversified funds and poorly-diversified portfolios ofstocks and low stock market participation (Polkovnichenko (2005)) and preference for skewness3 e, Strub and Zariphopoulou:
Forward Rank-Dependent Performance Criteria (Barberis and Huang (2008), Dimmock et al. (2018)).Solving portfolio optimization problems under rank-dependent utility preferences is difficultbecause such problems are both time-inconsistent and non-concave due to the probabilitydistortion. The difficulty of non-concavity was overcome by the quantile approach developedin Jin and Zhou (2008), Carlier and Dana (2011), He and Zhou (2011, 2016) and Xu (2016). Ageneral solution for a rank-dependend utility maximization problem in a complete market wasderived in Xia and Zhou (2016) and its effects on optimal investment decisions were extensivelystudied in He et al. (2017, 2018). On the other hand, it remains an open problem to solveportfolio optimization problems under rank-dependent utility in general incomplete markets,where one can not apply the martingale approach and time-inconsistency thus becomes a realchallenge.The difficulties in developing forward rank-dependent criteria are both conceptual and tech-nical. Conceptually, it is not clear what could replace the martingality/supermartingality re-quirements given that, in the classical setting, the DPP fails. Furthermore, there is not evena notion of (super)martingale under probability distortion. In addition, it is not clear howtime-consistency could be incorporated, if at all. From the technical point of view, challengesarise due to the fact that probability distortions are not amenable to infinitesimal stochasticcalculus, which plays a key role in deriving the forward stochastic PDE.We address these difficulties by first proposing two distinct definitions for a pair of processes, (cid:0) ( u t ( x )) t > , ( w s,t ( p )) s
1) combines in a very transparent way the market condition with the investor’s attitude.The latter can be thought as objective ( γ = 1) , pessimistic ( γ <
1) or optimistic ( γ > . As a corollary to the above results, we obtain that the distortion process of any forwardrank-dependent satisfies the monotonicity condition of Jin and Zhou (2008), cf. Assumption4.1 and the discussion in Section 6.2 therein. This implies in particular that the optimal wealthprocess is strictly decreasing as a function of the pricing kernel. An interesting analogy is a resultof Xia and Zhou (2016) showing that the Jin-Zhou monotonicity condition is also automatically5 e, Strub and Zariphopoulou:
Forward Rank-Dependent Performance Criteria satisfied for a representative agent of an Arrow-Debreu economy. In other words, we have thatthe monotonicity condition of Jin and Zhou (2008) is satisfied if the market is exogenously givenand the preferences are endogenously determined through the framework of forward criteria,or if the preferences are exogenously given and the pricing kernel is endogenously determinedthrough an equilibrium condition.The third main result is the actual construction of forward rank-dependent criteria. In thedegenerate case (3), it follows easily that u t ( x ) = u ( x ) , t > , for the optimal investment inall risky assets is always zero. In the non-degenerate case (2), we establish a direct equivalencewith deterministic, time-monotone forward criteria in the absence of probability distortions.Specifically, for a given γ, we introduce a new measure, the γ - distorted measure and a related distorted market with modified risk premium ˜ λ t := γλ t (see Subsection 5.1). As we explainlater on, the motivation for considering these measure and market variations comes from theform of the optimal wealth process for the non-degenerate case.In the distorted market, we in turn recall the standard (no probability distortion) time-monotone forward criterion, denoted by U t ( x ) . As established in Musiela and Zariphopoulou(2010a), it is given by U t ( x ) = v ( x, R ts k ˜ λ r k dr ) , with the function v ( x, t ) satisfying v t = v x v xx . Herein, we establish that u t ( x ) = U t ( x ) = v (cid:18) x, Z ts γ k λ r k dr (cid:19) . (4)In other words, the utility process u t ( x ) of the forward rank-dependent criterion in the originalmarket corresponds to a deterministic, time-monotone forward criterion in a pseudo-marketwith modified risk premia and vice-versa.If the investor is objective ( γ = 1) , there is no probability distortion and, as a result, thetwo markets become identical and the two criteria coincide, u t ( x ) = U t ( x ) . For optimisticinvestors ( γ > , however, the time-monotonicity of the function v ( x, t ) results in a morepronounced effect on how the forward rank-dependent utility decays with time. Specifically,the higher the optimism (higher γ ) , the larger the time-decay in the utility criterion, reflectinga higher loss of subjectively viewed better opportunities. The opposite behavior is observedfor pessimistic investors ( γ <
1) where the time-decay is slower, since the market opportunitieslook subjectively worse. Finally, the limiting case γ = 0 corresponds to a subjectively worthlessdistorted market. The latter yields U t ( x ) = U ( x ) = v ( x, , and in turn (4) implies that u t ( x ) = u ( x ) , t > . In addition to the construction approach, the equivalence established in Theorem 14 yieldsexplicit formulae for the optimal wealth and investment policies under forward probability dis-tortions by using the analogous formulae under deterministic, time-monotone forward criteria.As mentioned earlier, our construction of time-consistent criteria even in the presence ofprobability distortion prompts us to revisit the classical (backward) setting and investigate6 e, Strub and Zariphopoulou:
Forward Rank-Dependent Performance Criteria if and how our findings can be used to build time-consistent policies therein. The dynamicutility approach developed in Karnam et al. (2017) seems suitable to this end. This approachbuilds on the observation that the time-inconsistency of stochastic optimization problems ispartially due to the following restriction: The utility functional determining the objective atan intermediate time is essentially the same as the utility functional at initial time modulusconditioning on the filtration. The dynamic utility approach relaxes this restriction and allowsthe intermediate utility functional to vary more freely so that the DPP holds. In a recent workclosely related to this paper, Ma et al. (2018) introduce a dynamic distortion function . Thisleads to a distorted conditional expectation which is time-consistent in the sense that the tower-property holds. In their setting, an Itˆo process is given and fixed and the dynamic distortionfunction is then constructed for this particular process. Since the construction depends on thedrift and volatility parameters of the Itˆo process, their results are not directly applicable toour setting, where we consider an investment problem and the state process is not a priorigiven, but instead controlled by the investment policy. Herein, we extend the construction ofdynamic distortion functions to controlled processes for the problem of rank-dependent utilitymaximization in a financial market with determinstic coefficients. We find that constructinga dynamic utility which is restricted to remain in the class of RDU preference functionals ispossible if and only if the initial probability distortion function belongs to the family introducedin Wang (2000).Studying time-inconsistency induced by distorting probabilities is one of the remaining openchallenges for the psychology of tail events identified in the review article Barberis (2013). Wecontribute to this research direction by developing a new class of risk preferences and showingthat investment under probability can be time-consistent. Furthermore, we fully characterizethe conditions under which this is possible, namely if and only if the marginal probabilitydistortion equals to a normalized power of the quantile function of the pricing kernel.The paper is organized as follows. In Section 2, we describe the model and review the mainresults for the classical rank-dependent utility. In Section 3, we introduce the definitions ofthe forward rank-dependent performance criteria and establish their equivalence. We continuein Section 4, where we derive the necessary conditions for a distortion probability process tobelong to a forward rank-dependent pair. In Section 5, we establish the connection with thedeterministic, time-monotone forward criteria, the form of the optimal wealth and portfolioprocesses and provide examples. In Section 6, we relate our results to dynamic utility approachand show that constructing a dynamic utility restricted to remain within the class of RDUpreference functionals is possible if and only if the initial distortion function belongs to the classintroduced in Wang (2000). We conclude in Section 7. To ease the presentation, we delegate allproofs in an Appendix. 7 e, Strub and Zariphopoulou:
Forward Rank-Dependent Performance Criteria
We start with the description of the market model and a review of the main concepts andresults for rank-dependent utilities.The financial market consists of one risk-free and N risky assets. The price of the i th riskyasset solves dS it = S it µ it dt + N X j =1 σ ijt dW jt ! , t > , (5)with S i = s i > i = 1 , . . . , N . The process W = ( W t ) t > is an N -dimensional Brownianmotion on a filtered probability space (Ω , F , F , P ) satisfying the usual conditions and where F = ( F t ) t > is the completed filtration generated by W .The drift and volatility coefficients are assumed to be deterministic functions that satisfy R t | µ is | ds < ∞ and R t ( σ ijs ) ds < ∞ , t > i, j = 1 , . . . , N . We denote the volatility matrixby σ t := (cid:0) σ ijt (cid:1) N × N . We assume that σ t is invertible for all t > , to ensure that the market isarbitrage free and complete. We also define the market price of risk process, λ t := σ − t µ t (6)and assume that λ t > , t > t >
0, we consider the (unique) pricing kernel ρ t = exp (cid:18) − Z t k λ r k dr − Z T λ ′ r dW r (cid:19) . (7)For 0 < s t , we further define ρ s,t := ρ t ρ s = exp (cid:18) − Z ts k λ r k dr − Z ts λ ′ r dW r (cid:19) . (8)We also denote the cumulative distribution function of ρ t and ρ s,t by F ρt and F ρs,t respectively.We stress, and this will be also discussed later on, that while we pre-assume that the marketcoefficients are deterministic processes, we do not pre-specify their values. This is in contrastwith the classical setting where the full model (or a plausible family of models) needs to bedetermined at initial time and for the entire trading horizon, and thus the exact dynamics of µ it and σ ijt , i, j = 1 , . . . , N, have to be a priori known.The agent starts at t = 0 and trades between the riskless and the risky assets, using a self-financing trading policy ( π t , π t ) t > , where π t = π t ( ω ; x ) denotes the allocation in the risklessasset and the vector π t := (cid:0) π t , π t , ..., π Nt (cid:1) , with π it = π it ( ω ; x ) , i = 1 , . . . , N, representing theamount invested, at time t, in the risky asset i. Strategies are allowed to depend on the initialwealth x > ω ∈ Ω. We usually drop the ω and x argument whenever8 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria the context is clear. In turn, the wealth process X = ( X x,πt ) t > solves the stochastic differentialequation dX x,πt = π ′ t µ t dt + π ′ t σ t dW t , t > , (9)with X x,π = x. For notational simplicity, we will often write X π instead of X x,π . The set of admissible strategies is defined as A := (cid:26) π (cid:12)(cid:12) π t is F -progressively measurable,with Z t k π s k ds < ∞ and X x,πt > , for t > , x > (cid:27) . (10)For a given time, t , and an admissible policy, e π, we also introduce A ( e π, t ) := { π ∈ A| π s ≡ e π s , s ∈ [0 , t ] } , (11)namely, the set of admissible strategies which coincide with this specific policy in [0 , t ]. To ease the presentation and build motivation for the upcoming analysis, we start with a briefoverview of the rank-dependent utilities and the main results on portfolio optimization undersuch preferences for the market considered herein.The rank-dependent utility value of a prospect X is defined as V ( X ) := Z ∞ u ( ξ ) d ( − w (1 − F X ( ξ ))) , (12)where u is a utility function and w is a probability distortion function , cf. Quiggin (1982, 1993)and Schmeidler (1989).We assume that u and w belong to the sets U and W , introduced next. Definition 1.
Let U be the set of all utility functions u : [0 , ∞ ) → R , with u being strictlyincreasing, strictly concave, twice continuously differentiable in (0 , ∞ ) , and satisfying the Inadaconditions lim x ↓ u ′ ( x ) = ∞ and lim x ↑∞ u ′ ( x ) = 0 .Let W be the set of probability distortion functions w : [0 , → [0 , that are continuouslydifferentiable, strictly increasing and satisfying w (0) = 0 and w (1) = 1 . At initial time t = 0, an agent chooses her investment horizon T > , the dynamics in (5)for [0 , T ], together with u ∈ U and w ∈ W . She then solves the portfolio optimization problem v ( x,
0) = sup π ∈A T V ( X πT ) (13)9 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria with X πs , s ∈ [0 , T ] , solving (9) and X π = x, V is given in (12), and A T is defined similarly to A in (10), up to horizon T .This problem has been studied by various authors; see, among others, Carlier and Dana(2011), Xia and Zhou (2016), Xu (2016) or He et al. (2017, 2018). Fundamental difficulties arisefrom the time-inconsistency which stems from the probability distortion. Consequently, keyelements in stochastic optimization, like the Dynamic Programming Principle, the Hamilton-Jacobi-Bellman (HJB) equation, the martingality of the value function process along an opti-mum process and others, are lost.The analysis of (13) has been carried out using a well known reformulation to a staticproblem and the quantile method developed in Jin and Zhou (2008), Carlier and Dana (2011),He and Zhou (2011, 2016) and Xu (2016). Specifically, because the market is complete, any F T -measurable prospect X that satisfies the budget constraint E [ ρ T X ] = x can be replicatedby a self-financing policy. In turn, problem (13) reduces tosup X V ( X ) with E ( ρ T X ) x, X > , X ∈ F T . (14)One of the main steps in its solution are the specification of the “optimal” Lagrange multiplier,the construction of the terminal optimal wealth and the characterization of the optimal policythrough martingale representation results. The rank-dependent case, however, is considerablyharder due, from the one hand, the joint nonlinearities (risk preferences and non-linear av-eraging) in criterion (12) and, from the other, the non-concavity due to the presence of theprobability distortion.A very important feature in the RDU family of preferences is that, for certain choices of theprobability distortion function w , the optimal investment in the risky assets turns out to bezero, even if the market price of risk is not zero. The optimal wealth then remains unchanged(recall that interest rate is taken to be zero). We will be referring to this as a “degenerateoptimal investment case”. Note that this is in direct contrast with the classical setting where arisk averse agent would always invest in a worthy (non-zero risk premium) market.Central results on the optimal investment case were derived in Xia and Zhou (2016) andXu (2016) (see also Carlier and Dana (2011)) and are stated next. Theorem 2.
Let u ∈ U and w ∈ W . If there exists an optimal wealth to (14), it is given by X ∗ T = ( u ′ ) − (cid:16) λ ∗ ˆ N ′ (1 − w ( F ρT ( ρ T ))) (cid:17) , (15) where ˆ N is the concave envelope of N ( z ) := − Z w − (1 − z )0 ( F ρT ) − ( t ) dt, z ∈ [0 , . (16) and the Lagrangian multiplier λ ∗ > is determined by E [ ρ T X ∗ T ] = x . e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria
We also recall that if, in addition, the so called
Jin-Zhou monotonicity condition holds,namely, if the function f : [0 , → R + , defined as f ( p ) := ( F ρT ) − ( p ) w ′ ( p ) (17)is nondecreasing, then equality (15) simplifies to X ∗ T = ( u ′ ) − (cid:18) λ ∗ ρ T w ′ ( F ρT ( ρ T )) (cid:19) , (18)see Remark 3.4 in Xia and Zhou (2016) or Jin and Zhou (2008).Further results for problem (14) were derived in Xia and Zhou (2016), where it was shownthat if, for each λ > , the inequality E h ρ T ( u ′ ) − (cid:16) λ ˆ N ′ (1 − w ( F ρT ( ρ T ))) (cid:17)i < ∞ holds, with ˆ N ′ as in (16), then an optimal solution exists and is of form (15).In addition, the author in Xu (2016) showed that the existence of a non-degenerate optimalinvestment policy is equivalent to the existence of a Lagrangian multiplier λ ∗ and that, in thiscase, the terminal optimal wealth is as given in Theorem 2. We introduce the concept of forward performance criteria in the framework of rank-dependentpreferences. We first review the definition of the forward performance criterion (slightly modifiedfor the setting and notation herein); see, among others, Musiela and Zariphopoulou (2006, 2009,2010a). We then discuss the various difficulties in extending this concept when probabilitydistortions are incorporated.
Definition 3. An F -adapted process ( U t ) t > is a forward performance criterion ifi) for any t > and fixed ω ∈ Ω , U t ∈ U , ii) for any π ∈ A , s t and x > E [ U t ( X x,πt ) | F s ] U s ( X x,πs ) , (19) iii) there exists π ∗ ∈ A such that, for any s t, and x > , E h U t (cid:16) X x,π ∗ t (cid:17)(cid:12)(cid:12)(cid:12) F s i = U s (cid:0) X x,π ∗ s (cid:1) . (20)11 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria
The above definition was directly motivated by the DPP, a key feature in the classicalstochastic optimization, which yields the above supermartingality and martingality properties ofthe value function process along an admissible and an optimal policy, respectively. Furthermore,directly embedded in this fundamental connection between DPP and (19) and (20), is the time-consistency of the optimal policies.Once, however, probability distortions are incorporated, none of these features exist in theclassical rank-dependent case, as we discussed in the previous section. Indeed, the DPP doesnot hold and, naturally, time-inconsistency arises. Furthermore, there is no general notion ofsupermartingality and martingality under probability distortions, and thus it is not clear whatthe analogues of (19) and (20) are . In other words, we lack the deep connection among the DPP,the martingality/supermartingality of the value function process, and the time-consistency ofthe optimal policies, which is the cornerstone in the expected utility paradigm. Thus, it is notat all clear how to define the forward rank-dependent performance criteria. We address thesedifficulties in two steps.We first propose a definition of forward rank-dependent criteria by directly imitating re-quirements (19) and (20). Specifically, we propose (21) and (22), respectively, where we use(conditional) distorted probabilities instead of the regular ones. This definition is a natural,direct analogue to Definition 3, as it is built on the preservation of value along an optimalpolicy and its decay along a suboptimal one. The novel element in Definition 4 is that we seeka pair of processes (cid:0) ( u t ) t > , ( w s,t ) s
A pair of deterministic processes (cid:0) ( u t ) t > , ( w s,t ) s We now present an alternative definition. Definition 6. Let ( u t ) t > and ( w s,t ) s A pair of deterministic functions (cid:0) ( u t ) t > , ( w s,t ) s We present one of the main results herein, deriving necessary conditions for a deterministic prob-ability distortion process, w s,t , to belong to a forward rank-dependent pair (cid:0) ( u t ) t > , ( w s,t ) s Let the pair (cid:0) ( u t ) t > , ( w s,t ) s If the forward probability distortion is given by (25), then w s,t ( p ) = Φ Φ − ( p ) + ( γ − sZ ts k λ r k dr , (27) where ( λ t ) t > is the market price of risk (cf. (6)). The following result yields that it is necessary to allow the family of probability distor-tion functions of a forward rank-dependent performance process to depend on both the initialand terminal time. Otherwise, forward rank-dependent criteria reduce to the case without anyprobability distortion. Corollary 10. Let (cid:0) ( u t ) t , ( w s,t ) s Proposition 11. Let the pair (cid:0) ( u t ) t > , ( w s,t ) s Let V, V , V be RDU representations as in (12) with utility functions u, u , u and distortion functions w, w , w , respectively. Then,i) V is said to be pessimistic if for any X with bounded support, ∆ w ( X ) ≥ .ii) V is said to be more pessimistic than V if for any X with bounded support, ∆ w ( X ) > ∆ w ( X ) . The following proposition shows how the distortion parameter γ reflects the investor’s atti-tude as objective ( γ = 1) , pessimistic ( γ < 1) or optimistic ( γ > . Proposition 13. Let V, V , V be RDU representations as in (12) with utility functions u, u , u and distortion functions w, w , w given by (25) with distortion parameters γ, γ , γ respectively.Then the following holds:i) V is pessimistic if and only if γ .ii) V is more pessimistic than V if and only if γ γ . While most commonly used probability distortion functions, such as the ones introducedin Tversky and Kahneman (1992), Tversky and Fox (1995) or Prelec (1998), do not satisfythe Jin-Zhou monotonicity condition when paired with a lognormal pricing kernel, the proofof Theorem 8 shows that an endogenously determined probability distortion function of aforward rank-dependent performance criterion automatically satisfies this condition. In otherwords, while general classical (backward) RDU optimization problems are typically hard tosolve and rely on concavification techniques due to joint nonlinearities of the risk preferencesand non-linear averaging, the endogenous determination of the probability distortion by meansof forward criteria provides additional structure in terms of the Jin-Zhou monotonicity. This, inturn, leads to a simpler expression for the optimal wealth process as described in Proposition11. Interestingly, Xia and Zhou (2016) also find that the Jin-Zhou monotonicity condition isautomatically satisfied for a representative agent when it is the pricing kernel which is endoge-nously determined through an equilibrium condition of an Arrow-Debreu economy.Finally, we comment on the form of the optimal wealth process as it plays a pivotal role indeveloping the upcoming construction approach. In the degenerate case, we easily deduce that17 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria X ∗ t = x, t > . Note that the no participation effect occurs even if λ t > , t > , as assumedherein.For the non-degenerate case, we see that X ∗ t takes a form that resembles the one in theclassical setting but under a different measure, as manifested by the term E (cid:2) ρ − γt (cid:3) ρ γt in thesecond equality in (28). This motivated us to introduce a new measure, which in turn guidedus to develop a connection with the existing deterministic, time-monotone forward criteria inan auxiliary market. We present these results in the next section. This section contains the main result herein. It provides a direct connection between forwardrank-dependent performance criteria (cid:0) ( u t ) t > , ( w s,t ) s 0, we let P γ be the unique probability measure on (Ω , F ) satisfying,for each t > , d P γ d P (cid:12)(cid:12)(cid:12)(cid:12) F t = ρ − γt E [ ρ − γt ] . (30)Such a probability measure exists, and is unique and equivalent to P on (Ω , F t ) , for each t > P γ the γ -distorted probability measure , see alsoMa et al. (2018).In turn, the price processes (cf. (5)) solve dS t = σ t S t ( λ γ,t dt + dW γ,t ) , t > , (31)where λ γ,t := γλ t (32)and W γ,t := (1 − γ ) Z t λ s ds + W t (33)is a Brownian motion under P γ .We now consider an auxiliary market consisting of the riskless bond (with zero interestrate) and N stocks whose prices evolve as in (31) above. We will refer to this as the γ -distortedmarket. It is complete and its pricing kernel, denoted by ρ γ,t , is given by ρ γ,t = ρ t d P d P γ = ρ γt E [ ρ − γt ] . (34)18 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria In this auxiliary market, we recall the associated time-monotone forward performance criteria,denoted by U t ( x ) . It is given by U t ( x ) = v ( x, A γ,t ) with A γ,t := Z t k λ γ,s k ds. (35)The function v ( x, t ) solves, for x > , t > ,v t = 12 v x v xx , and v ( x, 0) must be of the form ( v ′ ) − ( x, 0) = R ∞ + x − y µ ( dy ) , where µ is a positive finite Borelmeasure. It also holds that v ( x, t ) = − Z t e − h − ( x,s )+ s ds + Z x e − h − ( z, dz, (36)with h ( z, t ) , z ∈ R , t > , given by h ( z, t ) := Z ∞ + e zy − y t µ ( dy ) . (37)We refer the reader to Musiela and Zariphopoulou (2010a) for an extensive exposition of theseresults as well as detailed assumptions on the underlying measure µ. We are now ready to present the main result herein, which connects the forward rank-dependent criteria in the original market with the deterministic, time-monotone forward criteriain the γ -distorted market and provides a construction method for forward rank-dependentperformance criteria. Theorem 14. Let γ ≥ . If ( u t ) t ≥ is a deterministic, time-monotone forward performancecriterion in the γ -distorted market and the family of probability distortions ( w s,t ) ≤ s 0) = R ∞ + x − y µ ( dy ) . Let h ( z, t ) and v ( x, t ) be given by (37) and (36). Then, the pair (cid:0) ( u t ) t > , ( w s,t ) s We stress that, while the necessary conditions on the probability distortion function inTheorem 8 have been established independently of the utility function process ( u t ) t > , bothprocesses ( u t , w s,t ) depend on the distortion parameter γ as (38) indicates. Indeed, γ > both as a parameter in the probability distortion function and in the rescaledtime argument for the utility function,through the process A γ,t .As γ ↓ , then lim γ ↓ A γ,t = 0 , for all t > , and in turn u t ( x ) = u ( x ) = v ( x, . This isexpected, as when γ = 0 , the risky asset prices in the γ − distorted market become martingales(cf. (31) and thus no participation is expected. Indeed, the γ -distorted measure P γ coincideswith the risk-neutral measure when γ = 0. Proposition 15. Let γ > be the investor’s distortion parameter and h ( z, t ) as in (37).Then, the associated optimal wealth, ( X ∗ t ) t > and investment policy ( π ∗ t ) t > corresponding to theforward rank-dependent performance criterion as constructed in (38) are given, respectively, by X ∗ t = h (cid:18) h − ( x, 0) + γ Z t k λ s k ds + γ Z t λ s dW s , γ Z t k λ s k ds (cid:19) (39) and π ∗ t = γσ − t λ t h x (cid:18) h − ( x, 0) + γ Z t k λ s k ds + γ Z t λ s dW s , γ Z t k λ s k ds (cid:19) . (40) ii) Let γ = 0 . Then, X ∗ t = x and π ∗ t = 0 , for all t > . We remind the reader that (39) and (40) offer an alternative expression for the optimalwealth and policy, already derived with different arguments in (28) and (29).Next, we provide examples where the underlying measure µ is a single Dirac or sum of twoDirac functions. Example 16. i) Let u ( x ) = − α x − α , α = 1 . Equivalently, µ ( dy ) = δ /α and, in turn, h ( x, t ) = e xα − α t and v ( x, t ) = − α x − α e (1 − α ) t . Let γ > . Then, the pair (cid:0) ( u t ) t > , ( w s,t ) s Let u ( x ) = − θ θ (1+ θ ) (cid:0) √ x + 1 − (cid:1) θ (cid:0) θ √ x + 1 + 1 (cid:1) , < θ < .Equivalently, the underlying measure is µ ( dy ) = δ − θ + δ − θ and therefore h ( x, t ) = e − θ x − 12 1(1 − θ )2 t + e − θ x − − θ )2 t and v ( x, t ) = 2 − θ θ (1 + θ ) e t (cid:16) − − θ + − θ )2 (cid:17) (cid:18)q xe − t (1 − θ )2 + 1 − (cid:19) θ (cid:18) θ q xe − t (1 − θ )2 + 1 + 1 (cid:19) Let γ > . Then, the pair (cid:0) ( u t ) t > , ( w s,t ) s Motivated by the construction of time-consistent rank-dependent criteria we here revisit theclassical (backward) RDU optimization problem. Specifically, we follow the dynamic utilityapproach developed in Karnam et al. (2017) and explore whether it is possible to derive a familyof dynamic RDU optimization problems under which the initial investment policy remainsoptimal over time. In a recent work related to this paper, Ma et al. (2018) utilize this approachto derive a time-consistent conditional expectation under probability distortion.We emphasize that, for this section only, we deviate from the theme of forward criteria inthat we consider a classical rank-dependent utility maximization problem of the formmax π ∈A Z ∞ u ,T ( ξ ) d (cid:0) − w ,T (cid:0) − F X πT ( ξ ) (cid:1)(cid:1) (45)with dX x,πr = π ′ r µ r dr + π ′ r σ r dW r , r ∈ [0 , T ] and X x,π = x > T > u ,T ∈ U and probability distortion function w ,T ∈ W . We assume that thereexists an optimal policy π ∗ to problem (45) and make the following definition of dynamicrank-dependent utility processes . Definition 18. A family of utility functions u t,T ∈ U , t ∈ (0 , T ) , and probability distortionfunctions w t,T ∈ W , t ∈ (0 , T ) is called a dynamic rank-dependent utility process for u ,T ∈ U and w ,T ∈ W over the time horizon T > if lim t ց u t,T = u ,T , lim t ց w t,T = w ,T and theoptimal policy π ∗ for (45) also solves, for any t ∈ (0 , T ) , max π ∈A ( π ∗ ,t ) Z ∞ u t,T ( ξ ) d (cid:16) − w t,T (cid:16) − F X x,πT |F t ( ξ ) (cid:17)(cid:17) (46) with dX x,πr = π ′ r µ r dr + π ′ r σ r dW r , r ∈ [0 , T ] and X x,π = x > . Our definition of dynamic rank-dependent utility processes relies on the existence of anoptimal policy for the initial problem (45). Karnam et al. (2017) do not rely on this assumptionand being able to determine the value of a time-inconsistent stochastic control problem withoutthe assumption of an optimal control is indeed one of their main contributions. However, forour specific problem of rank-dependent utility maximization in a complete, continuous-timefinancial market with deterministic coefficients, conditions for the existence of an optimal policyare well known and not restrictive, cf. Xia and Zhou (2016), Xu (2016) or He et al. (2017).We also deviate from Karnam et al. (2017) in that we restrict the objective of the dynamicfamily of problems (46) to belong to the same class of preferences as the initial problem (45),namely rank-dependent utility preferences. In Karnam et al. (2017) on the other hand, theobjective is allowed to vary more freely. However, we believe that within the dynamic utilityapproach, it is an interesting mathematical and economic question whether one is able toconstruct a dynamic utility belonging to the same class of preference functionals as the initial22 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria preferences and so this is exactly the question we want to address. There have been some positiveresults in this regard for generally time-inconsistent mean-risk portfolio optimization problems;see Cui et al. (2012) for the mean-variance and Strub et al. (2018) for the mean-CVaR problem.Ma et al. (2018) introduce the notion of a dynamic distortion function under which thedistorted, nonlinear conditional expectation is time-consistent, in the sense that the tower-property holds. Specifically, they consider an Itˆo process described by the stochastic differentialequation Y t = y + Z t b ( s, Y s ) ds + Z t σ ( s, Y s ) dW s , t T (47)and a given family of probability distortion functions ( w ,t ) t T , where w ,t applies over [0 , t ].Under technical conditions, they are able to derive a family of (random) probability distortionfunctions ( w s,t ) s t T such that w s,t is F s -measurable, for any 0 s t T , and the tower-property E r,t [ g ( Y t )] = E r,s [ E s,t [ g ( Y t )]] , r s t T, holds for any continuous, bounded, increasing and nonnegative function g , where E s,t denotesthe nonlinear conditional expectation E s,t [ ξ ] = Z ∞ w s,t ( P [ ξ > x ]) dx. We remark that the tower-property plays an important role in the theory on dynamic riskmeasures, see, e.g., Bielecki et al. (2017) for a survey, and refer to Ma et al. (2018) for furtherapplications and discussions.The important difference between the portfolio optimization problem we study here and thesetting of Ma et al. (2018) is that the Itˆo process described through the stochastic differentialequation (47) is given and fixed. In particular, there is no control or investment policy. More-over, the construction of the family of probability distortion functions ( w s,t ) s t T in Ma et al.(2018) depends on the drift b and volatility σ in (47), cf. Theorem 5.2 therein. In our setting, onthe other hand, the drift and volatility of the wealth process are controlled by the investmentpolicy π .It follows as a corollary to our results on forward rank-dependent performance processesthat, when the utility function u t,T = u ,T for any t ∈ (0 , T ) and the optimal policy invests isnot degenerate, we can construct a dynamic rank-dependent utility process if and only if theprobability distortion function belongs to the class of Wang (2000). The following theorem showsthat this result remains valid even if one allows both the utility function u t,T and probabilitydistortion function w t,T to depend on the initial time of the investment. 23 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria Theorem 19. Consider a fixed time-horizon T , utility function u ,T ∈ U and probability dis-tortion function w ,T ∈ W , and suppose that the optimal solution to (45) exists. Depending onthe probability distortion function w ,T , we have the following two cases:i) If w ,T ( p ) > E (cid:20) ρ ,T n ρ ,T ( F ρ ,T ) − ( p ) o (cid:21) , p ∈ [0 , , (48) then a family of utility functions u t,T ∈ U , t ∈ (0 , T ) , together with a family of probabilitydistortion functions w t,T ∈ W , t ∈ (0 , T ) , is a dynamic rank-dependent utility process for u ,T and w ,T if and only if the family of probability distortion functions satisfies w t,T ( p ) > E (cid:20) ρ t,T n ρ t,T ( F ρt,T ) − ( p ) o (cid:21) , p ∈ [0 , , for any t ∈ [0 , T ) .ii) If (48) does not hold, then a family of utility functions u t,T ∈ U , t ∈ (0 , T ) and probabilitydistortion functions w t,T ∈ W , t ∈ (0 , T ) with lim t ց u t,T = u ,T and lim t ց w t,T = w ,T is adynamic rank-dependent utility process for u ,T and w ,T if and only if there is a deterministicprocess γ t > , t ∈ [0 , T ) , continuous at zero and such that w t,T ( p ) = 1 E h ρ − γ t t,T i Z p (cid:16)(cid:0) F ρt,T (cid:1) − ( q ) (cid:17) − γ t dq = Φ Φ − ( p ) + ( γ t − sZ Tt k λ r k dr (49) for any t ∈ [0 , T ) , and the measure of risk-aversion of the dynamic utility function satisfies − u ′′ t,T ( x ) u ′ t,T ( x ) = − γ t γ u ′′ ,T ( x ) u ′ ,T ( x ) (50) for any t ∈ [0 , T ) and x > . Theorem 19 shows in particular that, when there is some non-zero investment in the riskyasset, an extension of the construction of dynamic distortion functions of Ma et al. (2018)to controlled processes is in general only possible if the initial probability distortion functionbelongs to the family introduced in Wang (2000).In order to maintain time-consistency, the utility function and probability distortion functionmust be coordinated with each other at different times through the relationship (50). Thisdynamic constraint connects the risk-aversion of the dynamic utility function with the dynamicparameter of the probability distortion function. Recall from Proposition 13 that the distortionparameter γ t reflects the investor’s attitude as objective ( γ t = 1), pessimistic ( γ t < 1) oroptimistic ( γ t > e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria and less risk-averse if she becomes more pessimistic. Moreover, the relationship between risk-aversion and pessimism as reflected in the parameter γ t is linear.Note that, if the risk-aversion is time-invariant, then γ t = γ , t ∈ (0 , T ), implying thatthe effect of probability distortion (measured by the time parameter t ) thus must decay overtime at the order of √ T − t , as it follows from (49). In particular, the probability distortioneffect should disappear when the remaining time approaches zero. On the other hand, if theprobability distortion function is time-invariant, we must have γ t = 1 + ( γ − r R T k λ s k ds R Tt k λ s k ds , t ∈ (0 , T ). Therefore, the measure of absolute risk-aversion of the utility function must increaseat the order of 1 / √ T − t as t approaches T . We introduced the concept of forward rank-dependent performance processes and thereby ex-tended the study of forward performance process to settings involving probability distortions.Forward rank-dependent performance criteria are herein taken to be deterministic. This madethe problem tractable but also guided us in building a fundamental connection with determin-istic, time-monotone forward criteria in a related market.We provided two alternative definitions, in terms of time-consistency and performance valuepreservation, respectively. We, then, provided a complete characterization of the viable proba-bility distortion functions. Specifically, we showed that for the non-degenerate case (non-zerorisky allocation) the probability distortion function resembles the one introduced by Wang(2000) but modified appropriately to capture the market evolution. We also showed that itsatisfies the Jin-Zhou monotonicity condition.We further derived the optimal wealth process, which then motivated the introduction ofthe distorted probability measure. We in turn established the key result, namely, a one-to-one correspondence between forward rank-dependent performance processes and deterministic,time-monotone forward performance criteria in the auxiliary market (under the distorted mea-sure). This results then allows to build on earlier findings of Musiela and Zariphopoulou (2009,2010a) to characterize forward rank-dependent performance criteria and their optimal processes.Finally, we related our results with the dynamic utility approach of Karnam et al. (2017)and, specifically, the dynamic distortion function of Ma et al. (2018). While Ma et al. (2018)are able to construct a dynamic distortion function which is time-consistent in the sense thatthe tower-property holds for a general class of initial probability distortion functions and givenand fixed state process, our results show that, when the wealth processes is controlled by theinvestment policy and there is investment in the risky asset, then time-consistent investmentunder probability distortion is possible if and only if the probability distortion belongs to theclass of Wang (2000). 25 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria Extending the deterministic case to the stochastic one is by no means trivial, as there areboth conceptual and technical challenges. Indeed, if one would allow u t and/or w s,t to be F t -measurable, in direct analogy to the forward performance case, then the value of a prospectas specified in (12) would be a random variable. Simply taking the expectation of this randomvalue of the prospect seems ad hoc. In particular, it seems unreasonable that an agent distortingprobabilities to evaluate a prospect would subsequently apply a mere linear expectation toaverage the resulting value of the prospect.There are a number of possible directions for future research. First, one might consider for-ward cumulative prospect theory performance criteria, which incorporate two further behavioralphenomena, namely reference dependence and loss aversion. There is a rich and active literatureon how the reference point evolves in time, and the results herein indicate that the frameworkof forward preferences seems suitable to derive conditions under which a time-varying referencepoint does not lead to time-inconsistent investment policies.A second possible direction is to consider discrete-time rank-dependent forward criteria.Indeed, much of the research in behavioral finance and economics assumes a discrete-timesetting. Furthermore, considering discrete-time forward criteria already lead to valuable insightsif there is no probability distortion.Finally, it would be interesting to extend the framework of forward rank-dependent per-formance criteria beyond problems of portfolio selection. Possible problems could for examplecome from the areas of pricing and hedging, insurance, optimal contracting, real world optionsor in situations where there is competition between different agents. Acknowledgments We are very grateful to Xunyu Zhou for suggesting the topic and his valuable comments. A Appendix. Proofs A.1 Proof of Proposition 5 If w ( p ) = p, p ∈ [0 , 1] and u ∈ U , then for any prospect X , Z ∞ u ( ξ ) d (cid:0) − w (cid:0) − F X |F t ( ξ ) (cid:1)(cid:1) = E [ u ( X ) |F t ] . e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria A.2 Proof of Proposition 7 Suppose that (cid:0) ( u t ) t > , ( w s,t ) s 0. Let A and A ( π ∗ , s ) asin (10) and (11). Then, there would be a policy, say π ∈ A ( π ∗ , s ) , such that Z ∞ u t ( ξ ) d (cid:0) − w s,t (cid:0) − F X x,πt |F s ( ξ ) (cid:1)(cid:1) > Z ∞ u t ( ξ ) d (cid:16) − w s,t (cid:16) − F X x,π ∗ t |F s ( ξ ) (cid:17)(cid:17) = u s (cid:0) X x,π ∗ s (cid:1) = u s ( X x,πs ) , on a set A s ∈ F s with P [ A s ] > 0. This however contradicts iii) of Definition 4. It then followsthat the optimal value of (23) is given by u s (cid:0) X x,π ∗ s (cid:1) .Next, assume that the pair (cid:0) ( u t ) t > , ( w s,t ) s 0, the inequality Z ∞ u t ( ξ ) d (cid:0) − w s,t (cid:0) − F X x,πt |F s ( ξ ) (cid:1)(cid:1) u s ( X x,πs )holds. We argue by contradiction. Suppose that there exists policy ¯ π ∈ A , time t > s with 0 s < t , an x > 0, a set ˜ A s ∈ F s with P [ A s ] > ε > Z ∞ u t ( ξ ) d (cid:16) − w s,t (cid:16) − F X x, ¯ πt |F s ( ξ ; ω ) (cid:17)(cid:17) > u s ( X x, ¯ πs ( ω )) + ε, ω ∈ ˜ A s . Using results on classical rank-dependent utility maximization (cf. Theorem 2), the wealthprocess corresponding to the policy π ∗ is given by X y,π ∗ t = (( u ′ t ) − (cid:16) λ ∗ t ( y ) ˆ N ′ ,t (cid:0) − w ,t (cid:0) F ρ ,t ( ρ ,t ) (cid:1)(cid:1)(cid:17) , for any initial wealth y > 0, where ˆ N ,t is the concave envelope of N ,t ( z ) := − Z w − ,t (1 − z )0 ( F ρ ,t ) − ( r ) dr, z ∈ [0 , , and λ ∗ t ( y ) is such that E h ρ t X y,π ∗ t i = y .Note that the range of ˆ N ′ ,t (cid:0) − w ,t (cid:0) F ρ ,t ( ρ ,t ) (cid:1)(cid:1) does not depend on the initial wealth y > . Furthermore, using results from Jin and Zhou (2008) or He and Zhou (2011, 2016), we obtainthat λ ∗ ,t ( y ) = u ′ ( y ), which has range (0 , ∞ ) due to the Inada condition.Therefore, for any δ > 0, there exist a, y > A s := (cid:26) X x, ¯ πs ; ∈ [ a, a + δ , X y,π ∗ s ∈ [ a + δ , a + δ ] (cid:27) ∩ ˜ A s , e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria is such that A s ∈ F s and P [ A s ] > ϑ r ( ω ; y ) := π ∗ r ( ω ; y ) + (¯ π r ( ω ; x ) − π ∗ r ( ω ; y )) (cid:0) [ s, ∞ ) ( r ) × A s ( ω ) × y = y (cid:1) . We then have that ϑ ∈ A ( π ∗ , s ) , since X y ,π ∗ s > X x, ¯ πs on A s .In turn, for ω ∈ A s we obtain Z ∞ u t ( ξ ) d (cid:16) − w s,t (cid:16) − F X y ,ϑt |F s ( ξ ; ω ) (cid:17)(cid:17) > Z ∞ u t ( ξ ) d (cid:16) − w s,t (cid:16) − F X x, ¯ πt |F s ( ξ ; ω ) (cid:17)(cid:17) > u s ( X x, ¯ πs ( ω )) + ε > u s (cid:0) X y ,π ∗ s ( ω ) − δ (cid:1) + ε > u s (cid:0) X y ,π ∗ s ( ω ) (cid:1) = Z ∞ u t ( ξ ) d (cid:16) − w s,t (cid:16) − F X y ,π ∗ t |F s ( ξ ; ω ) (cid:17)(cid:17) , where the last inequality holds for small enough δ . This however contradicts the optimality of π ∗ and we easily conclude. A.3 Proof of Theorem 8 From Proposition 7 we have that (cid:0) ( u t ) t > , ( w s,t ) s 0. This becomes λ ∗ s,t (cid:16) ( u ′ s ) − (cid:16) λ ∗ ,s ( x ) ˆ N ′ ,s (cid:0) − w ,s (cid:0) F ρ ,s ( ρ ,s ) (cid:1)(cid:1)(cid:17)(cid:17) ˆ N ′ s,t (cid:0) − w s,t (cid:0) F ρs,t ( ρ s,t ) (cid:1)(cid:1) = λ ∗ ,t ( x ) ˆ N ′ ,t (cid:0) − w ,t (cid:0) F ρ ,t ( ρ ,t ) (cid:1)(cid:1) . (53)Next, we define the auxiliary functions h ,xs,t , h s,t , h ,xs,t : (0 , ∞ ) → (0 , ∞ ) by h ,xs,t ( y ) = λ ∗ s,t (cid:16) ( u ′ s ) − (cid:16) λ ∗ ,s ( x ) ˆ N ′ ,s (cid:0) − w ,s (cid:0) F ρ ,s ( y ) (cid:1)(cid:1)(cid:17)(cid:17) ,h s,t ( y ) = ˆ N ′ s,t (cid:0) − w s,t (cid:0) F ρs,t ( y ) (cid:1)(cid:1) ,h ,xs,t ( y ) = λ ∗ ,t ( x ) ˆ N ′ ,t (cid:0) − w ,t (cid:0) F ρ ,t ( y ) (cid:1)(cid:1) . Since ρ ,s and ρ s,t are independent with ρ ,s ρ s,t = ρ ,t we deduce that, for all y, z > ,h ,xs,t ( y ) h s,t ( z ) = h ,xs,t ( yz ) . (54)Next, suppose that there exist y , y with 0 < y < y and such that h ,xs,t ( y ) = h ,xs,t ( y ).Then, equality (54) together with the monotonicity of b N ′ imply that h ,xs,t , and in turn h ,xs,t and h s,t , are constant. Hence, ˆ N ′ s,t ( z ) = 1 , for all z ∈ [0 , N s,t ( z ) z − z ∈ [0 , N s,t in (52) and substituting x = w − s,t (1 − z ) yields that the aboveinequality is equivalent to (26). The same argument can be made if there exist z , z with0 < z < z and such that h s,t ( z ) = h s,t ( z ). Similarly, if there exist ξ , ξ with 0 < ξ < ξ and h ,xs,t ( ξ ) = h ,xs,t ( ξ ), then h ,xs,t ( ξ ) h s,t (1) = h ,xs,t ( ξ ) = h ,xs,t ( ξ ) = h ,xs,t ( ξ ) h s,t (1) and the statementfollows.Hence, there are two cases: either the family of probability distortion functions satisfies (26)or h ,xs,t is strictly decreasing and h s,t , h ,xs,t strictly increasing. On the other hand, the latter casein turn is equivalent to N s,t ( · ) being concave, that is, that the probability distortion w s,t satisfiesthe Jin-Zhou monotonicity condition, namely, that the function ( F ρs,t ) − ( · ) w ′ s,t ( · ) is nondecreasing.Next, we suppose that the family of probability distortion functions does satisfy the Jin-Zhou monotonicity condition. Because we have just shown that in this case N s,t is concave,and thus coincides with its concave envelope, a straightforward computation shows that thefunctions h ,xs,t , h s,t and h ,xs,t simplify to h ,xs,t ( y ) = λ ∗ s,t ( u ′ s ) − λ ∗ ,s ( x ) yw ′ ,s (cid:0) F ρ ,s ( y ) (cid:1) !! , e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria h s,t ( y ) = yw ′ s,t (cid:0) F ρs,t ( y ) (cid:1) and h ,xs,t ( y ) = λ ∗ ,t ( x ) yw ′ ,t (cid:0) F ρ ,t ( y ) (cid:1) . Taking y = 1 in (54) yields h ,xs,t ( z ) = h ,xs,t (1) h s,t ( z ) while taking z = 1 gives h ,xs,t ( y ) = h ,xs,t ( y ) h s,t (1). Combining the two gives h ,xs,t ( yz ) = h ,xs,t ( y ) h s,t ( z ) = h ,xs,t ( y ) h ,xs,t ( z ) h ,xs,t (1) h s,t (1) = h ,xs,t ( y ) h ,xs,t ( z ) h ,xs,t (1) . Next, we define g : R → R by g ( z ) := log (cid:0) h ,xs,t ( e z ) (cid:1) − log h ,xs,t (1). We deduce that g satisfiesCauchy’s functional equation g ( y + z ) = g ( y ) + g ( z ). Indeed, g ( y + z ) = log (cid:0) h ,xs,t (cid:0) e y + z (cid:1)(cid:1) − log h ,xs,t (1)= log h ,xs,t ( e y ) h ,xs,t ( e z ) h ,xs,t (1) ! − log h ,xs,t (1)= log (cid:0) h ,xs,t ( e y ) (cid:1) + log (cid:0) h ,xs,t ( e z ) (cid:1) − log (cid:0) h ,xs,t (1) (cid:1) − log h ,xs,t (1)= g ( y ) + g ( z ) , for any y, z ∈ R . Since g is continuous there must be a γ ∈ R such that g ( z ) = γz , z ∈ R . This,in turn, yields that for z > , h ,xs,t ( z ) = h ,xs,t (1) z γ . On the other hand, because h ,xs,t is strictly increasing when the family of probability distor-tion functions does satisfy the Jin-Zhou monotonicity condition, it must be that γ is positive.We, therefore, obtain that for p ∈ [0 , ,w ′ s,t ( p ) = λ ∗ s,t ( x ) h ,xs,t (1) (cid:16)(cid:0) F ρs,t (cid:1) − ( p ) (cid:17) − γ . Furthermore, since1 = Z w ′ s,t ( p ) dp = Z λ ∗ s,t ( x ) h ,xs,t (1) (cid:16)(cid:0) F ρs,t (cid:1) − ( p ) (cid:17) − γ dp = λ ∗ s,t ( x ) h ,xs,t (1) E (cid:2) ρ − γs,t (cid:3) , we have that h ,xs,t (1) = λ ∗ s,t ( x ) E (cid:2) ρ − γ ,t (cid:3) and in turn w s,t ( p ) = 1 E (cid:2) ρ − γs,t (cid:3) Z p (cid:16)(cid:0) F ρs,t (cid:1) − ( q ) (cid:17) − γ dq. Finally, since h s,t ( z ) = h ,xs,t ( z ) h ,xs,t (1) , we obtain (25) for any 0 s < t , using the same arguments asabove. 30 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria A.4 Proof of Corollary 10 For the first case of Theorem 8, the only γ > w s,t ( p ) = 1 E (cid:2) ρ − γs,t (cid:3) Z p (cid:16)(cid:0) F ρs,t (cid:1) − ( q ) (cid:17) − γ dq is independent of s, for every s ∈ [0 , t ) , is when γ = 1 and the assertion follows.The second case of Theorem 8 cannot happen when w s,t is independent of s . Indeed, when s goes to t , then the right-hand side of (26) converges to the mapping p { p> } . However,the distortion function w s,t must be continuous, and we conclude. A.5 Proof of Proposition 11 Case ii) follows immediately from the proof of Theorem 8 and thus we only need to showcase i). We first show that the Lagrangian multiplier satisfies λ ∗ s,t ( X ∗ s ) = u ′ s ( X ∗ s ). Indeed, fromTheorem 8 and the functional relation (53), we deduce that λ ∗ s,t (cid:16) ( u ′ s ) − (cid:0) λ ∗ ,s ( x ) E (cid:2) ρ − γ ,s (cid:3) ρ γ ,s (cid:1)(cid:17) E (cid:2) ρ − γs,t (cid:3) ρ γs,t = λ ∗ ,t ( x ) E (cid:2) ρ − γ ,t (cid:3) ρ γ ,t . Therefore, λ ∗ s,t (cid:16) ( u ′ s ) − (cid:0) u ′ ( x ) E (cid:2) ρ − γ ,s (cid:3) ρ γ ,s (cid:1)(cid:17) = u ′ ( x ) E (cid:2) ρ − γ ,s (cid:3) ρ γ ,s . From this, we conclude that λ ∗ s,t ( x ) = u ′ s ( x ), for all x > X ∗ t = ( u ′ t ) − λ ∗ ,t ( x ) ρ t w ′ ρ ,tt ( ρ t )) ! = ( u ′ t ) − (cid:0) u ′ ( x ) E (cid:2) ρ − γt (cid:3) ρ γt (cid:1) , t > , where we used the form of the Lagrangian multiplier determined above. From this, we deducethat X ∗ t = ( u ′ t ) − (cid:0) λ ∗ s,t ( X ∗ s ) E (cid:2) ρ − γs,t (cid:3) ρ γs,t (cid:1) = ( u ′ t ) − (cid:0) u ′ ( x ) E (cid:2) ρ − γt (cid:3) ρ γt (cid:1) , s < t. A.6 Proof of Proposition 13 Note that w ( p ) ≥ p for all p ∈ [0 , 1] if and only if γ > w ( p ) ≥ w ( p ) for all p ∈ [0 , 1] ifand only if γ ≥ γ . Thus, the assertion follows by Propositions 2.3 and 2.6 in Ghossoub and He(2018). 31 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria A.7 Proof of Theorem 14 We first prove the correspondence between forward rank-dependent performance processes anddeterministic, time-monotone forward performance processes under the distorted probabilitymeasure. We start with the converse direction.Let (cid:0) ( u t ) t > , ( w s,t ) s 0) is a deterministic, time-monotone forward performancecriterion in the γ -distorted market by Proposition 5 and Proposition 7.Now suppose that we are in case (25). Let P γ be given by (30) and denote the expectationunder it by E γ . According to Proposition 11, the wealth process X ∗ also solves the family ofexpected utility maximization problems generated by the optimal policy, say π ∗ ,max X E γ [ u t ( X )] (55)with E γ (cid:2) ˜ ρ s,t X (cid:12)(cid:12) F s (cid:3) = X ∗ s , X > , X ∈ F t . Moreover, we have that u s ( X ∗ s ) = Z ∞ u t ( ξ ) d (cid:0) − w s,t (cid:0) − F X ∗ t |F s ( ξ ) (cid:1)(cid:1) = Z ∞ u t ( ξ ) d − w s,t P ρ s,t u ′ t ( ξ ) u ′ s ( X ∗ s ) E (cid:2) ρ − γs,t (cid:3) ! /γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F s = Z ∞ u t (cid:16) ( u ′ t ) − (cid:0) u ′ s ( X ∗ s ) E (cid:2) ρ − γs,t (cid:3) y γ (cid:1)(cid:17) dw s,t (cid:0) F ρs,t ( y ) (cid:1) = E h u t (cid:16) ( u ′ t ) − (cid:0) u ′ s ( X ∗ s ) E (cid:2) ρ − γs,t (cid:3) ρ γs,t (cid:1)(cid:17) w ′ s,t (cid:0) F ρs,t (cid:0) ρ s,t (cid:1)(cid:1)(cid:12)(cid:12)(cid:12) F s i = E " u t (cid:16) ( u ′ t ) − (cid:0) u ′ s ( X ∗ s ) e ρ s,t (cid:1)(cid:17) ρ − γs,t E (cid:2) ρ − γs,t (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F s = E γ h u t (cid:16) ( u ′ t ) − (cid:0) u ′ s ( X ∗ s ) e ρ s,t (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) F s i = E γ [ u t ( X ∗ t ) | F s ] . (56)Therefore, for each fixed t > 0, we have that, from the one hand, u s corresponds to the valuefunction of the expected utility maximization problem under the distorted measure P γ , (55),with time horizon t and utility function u t , and, from the other, this policy π ∗ is optimal.Hence, E γ [ u t ( X πt ) | F s ] u s ( X πs ) for any admissible policy π with the same argument as inProposition 7. Thus, ( u t ) t > is a forward performance criterion in the γ -distorted market.To establish the other direction we work as follows. Let γ > u t ) t > be a de-terministic, time-monotone forward performance process in the γ -distorted market. Together32 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria with ˜ w s,t ( p ) ≡ p , p ∈ [0 , 1] for all 0 s < t , the pair (cid:0) ( u t ) t > , ( ˜ w s,t ) s Following the results in Musiela and Zariphopoulou (2010a), we deduce that the optimal wealthunder the time-monotone forward performance criteria in the γ -distorted market is given by X ∗ t = h (cid:18) h − (0 , t ) + Z t k λ γ,s k ds + Z t λ γ,s dW γ,s , Z t k λ γ,s k ds (cid:19) , with h as in (37). Using (32) and (33) we conclude. We similarly deduce (40). A.9 Proof of Theorem 19 Recall that the optimal solution tomax X Z ∞ u t,T ( ξ ) d ( − w t,T (1 − F X ( ξ )))s . t . E (cid:2) ρ t,T X |F t (cid:3) = E (cid:2) ρ t,T X ∗ |F t (cid:3) , X > , X is F T − measurable . is given by X ∗ ,t = ( u ′ t,T ) − (cid:16) λ ∗ t,T (cid:0) E (cid:2) ρ t,T X ∗ |F t (cid:3)(cid:1) ˆ N ′ t,T (cid:0) − w t,T (cid:0) F ρt,T ( ρ t,T ) (cid:1)(cid:1)(cid:17) , where ˆ N t,T is the concave envelope of (52) and λ ∗ t,T ( X t ) > E (cid:2) ρ t,T X ∗ ,t (cid:12)(cid:12) F t (cid:3) = E [ ρ T X ∗ |F t ]. Optimality of the initial optimal solution X , ∗ = X ∗ is thus maintained if and33 e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria only if λ ∗ t,T (cid:0) E (cid:2) ρ t,T X ∗ |F t (cid:3)(cid:1) ˆ N ′ t,T (cid:0) − w t,T (cid:0) F ρt,T ( ρ t,T ) (cid:1)(cid:1) = u ′ t,T (cid:16) ( u ′ ,T ) − (cid:16) λ ∗ ,T ( x ) ˆ N ′ ,T (cid:0) − w ,T (cid:0) F ρ ,T ( ρ ,T ) (cid:1)(cid:1)(cid:17)(cid:17) . (57)Similar to the proof of Theorem 8, we define g ,xt , g t , g ,xt : (0 , ∞ ) → (0 , ∞ ) by g ,xt ( y ) = λ ∗ t,T (cid:16) E h ρ t,T ( u ′ ,T ) − (cid:16) λ ∗ ,T ( x ) ˆ N ′ ,T (cid:0) − w ,T (cid:0) F ρ ,T ( yρ t,T ) (cid:1)(cid:1)(cid:17)i(cid:17) ,g t ( y ) = ˆ N ′ t,T (cid:0) − w t,T (cid:0) F ρt,T ( y ) (cid:1)(cid:1) ,g ,xt ( y ) = u ′ t,T (cid:16) ( u ′ ,T ) − (cid:16) λ ∗ ,T ( x ) ˆ N ′ ,T (cid:0) − w ,T (cid:0) F ρ ,T ( y ) (cid:1)(cid:1)(cid:17)(cid:17) . (58)By the independence of ρ ,t and ρ t,T and since ρ ,t ρ t,T = ρ ,T we have (57) if and only if g ,xt ( y ) g t ( z ) = g ,xt ( yz ) (59)for all y, z > 0. Since u ′ t,T is strictly decreasing for any t ∈ [0 , T ), we can make the sameargument as in the proof of Theorem 8 to conclude that this holds if and only if either w t,T ( p ) > E (cid:20) ρ t,T n ρ t,T ( F ρt,T ) − ( p ) o (cid:21) , p ∈ [0 , , for any t ∈ [0 , T ), or the family of probability distortions functions satisfies the Jin-Zhoumonotonicity condition. The first case proves the first part of the theorem.For the latter case, we first show the only if direction by following a similar line of argumentas in Theorem 8. If the probability distortion functions satisfy the Jin-Zhou monotonicitycondition, g ,xt , g t , g ,xt : (0 , ∞ ) → (0 , ∞ ) defined in (58) simplify to g ,xt ( y ) = λ ∗ t,T E " ρ t,T ( u ′ ,T ) − λ ∗ ,T ( x ) yρ t,T w ′ ,T (cid:0) F ρ ,T (cid:0) yρ t,T (cid:1)(cid:1) ! ,g t ( y ) = yw ′ t,T (cid:0) F ρt,T ( y ) (cid:1) ,g ,xt ( y ) = u ′ t,T ( u ′ ,T ) − λ ∗ ,T ( x ) yw ′ ,T (cid:0) F ρ ,T ( y ) (cid:1) !! . As in the proof of Theorem 8, we can show that g ,xt satisfies Cauchy’s functional equation andconclude that there is a γ t > g ,xt ( y ) = g ,xt (1) y γ t for all y > 0. Thus, by virtue of(59), g ,xt ( y ) y γ t = g ,xt (1) z γ t g t ( z ) = C x,t . e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria for some constant C x,t . From the definition of g t we obtain that for any t ∈ (0 , T ), w t,T ( p ) = 1 E h ρ − γ t t,T i Z p (cid:16)(cid:0) F ρt,T (cid:1) − ( q ) (cid:17) − γ t dq. By continuity of w t,T in t at zero, w ,T ( p ) = 1 E h ρ − γ ,T i Z p (cid:16)(cid:0) F ρ ,T (cid:1) − ( q ) (cid:17) − γ dq where γ = lim t ց γ t . From g ,xt ( y ) = g ,xt (1) y γ t we thus obtain that u ′ t,T (cid:16) ( u ′ ,T ) − (cid:16) λ ∗ ,T ( x ) E h ρ − γ ,T i y γ (cid:17)(cid:17) = g ,xt (1) y γ t . With the substitution z = (cid:0) u ′ ,T (cid:1) − (cid:16) λ ∗ ,T ( x ) E h ρ − γ ,T i y γ (cid:17) this becomes u ′ t,T ( z ) = g ,xt (1) (cid:16) λ ∗ ,T ( x ) E h ρ − γ ,T i(cid:17) γ t /γ (cid:0) u ′ ,T ( z ) (cid:1) γ t /γ . (60)Differentiating (60) with respect to z yields u ′′ t,T ( z ) = g ,xt (1) (cid:16) λ ∗ ,T ( x ) E h ρ − γ ,T i(cid:17) γ t /γ γ t γ (cid:0) u ′ ,T ( z ) (cid:1) γ t /γ − u ′′ ,T ( z ) . (61)Dividing (61) by (60) gives (50).For the if direction, we fix t ∈ (0 , T ) and first note that (50) is equivalent to ddz log (cid:0) u ′ t,T ( z ) (cid:1) = γ t γ ddz log (cid:0) u ′ ,T ( z ) (cid:1) and thus u ′ t,T ( z ) = e C (cid:0) u ′ ,T ( z ) (cid:1) γ t /γ for some constant e C ∈ R . From this we derive (cid:0) u ′ t,T (cid:1) − (cid:16) e Cz γ t (cid:17) = (cid:0) u ′ ,T (cid:1) − ( z γ ) , (62) z ∈ (0 , ∞ ). According to Theorem 2, the optimal solution to (46) is given by X ∗ ,t = ( u ′ t,T ) − (cid:16) λ ∗ t,T (cid:0) E (cid:2) ρ t,T X ∗ |F t (cid:3)(cid:1) E h ρ − γ t t,T i ρ γ t t,T (cid:17) , where λ ∗ t,T ( X t ) > E (cid:2) ρ t,T X ∗ ,t (cid:12)(cid:12) F t (cid:3) = E [ ρ T X ∗ |F t ]. From Theorem 4.1 in He et al.(2017) we furthermore have that E [ ρ T X ∗ |F t ] = G t,λ ∗ ( x ) ( ρ t ) for some strictly decreasing andthus invertible function G t,λ ∗ ( x ) . We define λ t,T : (0 , ∞ ) → (0 , ∞ ) by λ t,T ( ξ ) := e C (cid:0) λ ∗ ,T ( x ) (cid:1) γ t /γ E h ρ − γ ,T i γ t /γ E h ρ − γ t t,T i G − t,λ ∗ ( x ) ( ξ ) γ t . e, Strub and Zariphopoulou: Forward Rank-Dependent Performance Criteria Using the relation (62) we obtain( u ′ t,T ) − (cid:16) λ t,T (cid:0) E (cid:2) ρ t,T X ∗ |F t (cid:3)(cid:1) E h ρ − γ t t,T i ρ γ t t,T (cid:17) = ( u ′ t,T ) − (cid:18) e C (cid:0) λ ∗ ,T ( x ) (cid:1) γtγ E h ρ − γ ,T i γtγ ρ γ t ,T (cid:19) = ( u ′ ,T ) − (cid:16) λ ∗ ,T ( x ) E h ρ − γ ,T i ρ γ ,T (cid:17) = X ∗ . We in particular have that E (cid:20) ρ t,T ( u ′ t,T ) − (cid:16) λ t,T (cid:0) E (cid:2) ρ t,T X ∗ |F t (cid:3)(cid:1) E h ρ − γ t t,T i ρ γ t t,T (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = E (cid:2) ρ t,T X ∗ (cid:12)(cid:12) F t (cid:3) . By the uniqueness of the Lagrangian multiplier we conclude that λ t,T ( · ) = λ ∗ t,T ( · ) and therefore X ∗ ,t = X ∗ . References Angoshtari, B., Zariphopoulou, T., Zhou, X. 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